| L(s) = 1 | − 2·5-s + 9-s − 8·13-s + 12·17-s + 3·25-s + 12·29-s + 4·37-s − 12·41-s − 2·45-s + 49-s − 24·53-s + 28·61-s + 16·65-s − 8·73-s + 81-s − 24·85-s + 12·89-s − 32·97-s + 12·101-s + 4·109-s − 8·117-s + 14·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 2.21·13-s + 2.91·17-s + 3/5·25-s + 2.22·29-s + 0.657·37-s − 1.87·41-s − 0.298·45-s + 1/7·49-s − 3.29·53-s + 3.58·61-s + 1.98·65-s − 0.936·73-s + 1/9·81-s − 2.60·85-s + 1.27·89-s − 3.24·97-s + 1.19·101-s + 0.383·109-s − 0.739·117-s + 1.27·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474337603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474337603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.244873816754916675142815177713, −7.85604148921878259917464879318, −7.52869725497309556858557046810, −6.99473100991362891718403527162, −6.74469025095782946469111005872, −6.08311986208269073176654815515, −5.31154804232921291759313294728, −5.14033706479088381770575053688, −4.64347590023529327148780089392, −4.14525984671534409011710586104, −3.26877398752888167972289655677, −3.16591912200898921503339464095, −2.44845262549261294984946742912, −1.47447091118007102923277723016, −0.63276729269256935711871700309,
0.63276729269256935711871700309, 1.47447091118007102923277723016, 2.44845262549261294984946742912, 3.16591912200898921503339464095, 3.26877398752888167972289655677, 4.14525984671534409011710586104, 4.64347590023529327148780089392, 5.14033706479088381770575053688, 5.31154804232921291759313294728, 6.08311986208269073176654815515, 6.74469025095782946469111005872, 6.99473100991362891718403527162, 7.52869725497309556858557046810, 7.85604148921878259917464879318, 8.244873816754916675142815177713
